There are two main types of waves generated by earthquakes, body waves (P-waves, S-waves) and surface waves, which includes Love and Rayleigh waves. Surface waves are generated by the constructive interference of incident P and S -waves arriving at the free surface and propagating parallel to the surface. The amplitude of surface waves decreases with increasing depth and are affected by lateral variations in structure. Another property that surface waves exhibit is Dispersion, where the velocity of a wave on the surface is dependent on its frequency (or period). These constructive interference patterns travel in wave-trains with their own velocity called the group velocity, with different frequencies arriving at different times. Velocity normally increases with depth so longer period waves are faster because they sample deeper into the Earth where the velocities are higher. Short period waves behave just the opposite, they are slower because they travel through lower velocity material. Phase velocity is the culmination of a continuous spectrum of frequencies where each harmonic component has its own velocity (Lay and Wallace, 1995). Phase velocity is dependent on the source of the earthquake and requires a good knowledge of the moment tensor and depth for accurate measurement. Because surface waves only sample the upper few hundred kilometers of the crust and upper mantle, phase and group velocities can be used to determine upper crustal structure by looking at velocity changes with depth. In terms of my research, I use Rayleigh and Love group and phase velocities to do a one-dimesional inversion to look at the velocity structure around China's nuclear test site. By knowing what the dispersive characteristics are in a regional setting, one can understand how energy (seismic and explosive) travels through the upper crust. This is important for discriminating between a naturally occuring earthquakes and a nuclear test. I will try to illustrate these ideas as simplistic as possible. There are numerous books and papers on this subject, these are the "classic" books if you are interested in exploring the theory more: Aki and Richards (1980), Lay and Wallace (1995), Shearer (1999), Udias (1999). Here are some papers that pertain to Central Asian dispersion studies: Levshin and Ritzwoller (1995), Mahdi and Pavlis (1998), Pavlis and Mahdi (1996), Ritzwoller et al., (1998), Roecker, et al., (1993). These should keep you busy for a while.
Surface waves account for nearly all of the damage that occurs from the shaking associated with earthquakes. They can literally tear, split or in the case above in Niigata, Japan, knock over even so called "earthquake proof" buildings.
Here is a nice little table that relates the most common quantities used in describing seismic waves as they propagate.